
This ensures that \(F = V_n\)  is satisfied on the interface and gives a continuous expression for \(F\)  away from the interface. For the purpose of discretisation, the sign of \(\phi \)  is used to determine nodes inside the interface that need to be included in the biharmonic stencil. This discretisation results in a symmetric system of linear equations which is solved using LU decomposition. As such, the location of the interface does not need to be known explicitly, similar to the level set method itself. This velocity extension process is a variant of a thin plate spline in two dimensions [1]}. To illustrate the velocity extension process, we consider an example \(F\)  defined in the region \(\Omega (t)\) , shown in fig:VelocityExtension\((a)\) . The red line represented the fluid-fluid interface \(\partial \Omega (t)\) . fig:VelocityExtension\((b)\)  shows \(F\)  after the biharmonic equation (REF ) is solved. We see that this velocity extension process gives a differentiable expression for \(F\)  over the entire computational domain, which can be used to solve (REF ).
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